Invariant descriptive set theory / Su Gao.

Format:

Book

Author/Creator:

Gao, Su, 1968-

Contributor:

Hazel M. Hussong Fund.

Series:

Monographs and textbooks in pure and applied mathematics ; 293.

Pure and applied mathematics. Monographs and textbooks in pure and applied mathematics ; 293

Language:

English

Subjects (All):

Descriptive set theory.

Invariant sets.

Physical Description:

xiv, 383 pages : illustrations ; 25 cm.

Place of Publication:

Boca Raton : CRC Press, [2009]

Summary:

Exploring an active area of mathematics that studies the complexity of equivalence relations and classification problems, Invariant Descriptive Set Theory presents an introduction to the basic concepts, methods, and results of this theory. It brings together techniques from various areas of mathematics, such as algebra, topology, and logic, which have diverse applications to other fields.

After reviewing classical and effective descriptive set theory, the text studies Polish groups and their actions. It then covers Borel reducibility results on Borel, orbit, and general definable equivalence relations. The author also provides proofs for numerous fundamental results, such as the Glimm-Effros dichotomy, the Burgess trichotomy theorem, and the Hjorth turbulence theorem. The next part describes connections with the countable model theory of infinitary logic, along with Scott analysis and the isomorphism relation on natural classes of countable models, such as graphs, trees, and groups. The book concludes with applications to classification problems and many benchmark equivalence relations.

By illustrating the relevance of invariant descriptive set theory to other fields of mathematics, this self-contained book encourages readers to further explore this very active area of research.

Contents:

I Polish Group Actions 1

1.1 Polish spaces 4

1.2 The universal Urysohn space 8

1.3 Borel sets and Borel functions 13

1.4 Standard Borel spaces 18

1.5 The effective hierarchy 23

1.6 Analytic sets and [Sigma superscript 1 subscript 1] sets 29

1.7 Coanalytic sets and [Pi superscript 1 subscript 1] sets 33

1.8 The Gandy-Harrington topology 36

2 Polish Groups 39

2.1 Metrics on topological groups 40

2.2 Polish groups 44

2.3 Continuity of homomorphisms 51

2.4 The permutation group S[subscript infinity] 54

2.5 Universal Polish groups 59

2.6 The Graev metric groups 62

3 Polish Group Actions 71

3.1 Polish G-spaces 71

3.2 The Vaught transforms 75

3.3 Borel G-spaces 81

3.4 Orbit equivalence relations 85

3.5 Extensions of Polish group actions 89

3.6 The logic actions 92

4 Finer Polish Topologies 97

4.1 Strong Choquet spaces 97

4.2 Change of topology 102

4.3 Finer topologies on Polish G-spaces 105

4.4 Topological realization of Borel G-spaces 109

II Theory of Equivalence Relations 115

5 Borel Reducibility 117

5.1 Borel reductions 117

5.2 Faithful Borel reductions 121

5.3 Perfect set theorems for equivalence relations 124

5.4 Smooth equivalence relations 128

6 The Glimm-Effros Dichotomy 133

6.1 The equivalence relation E[subscript 0] 133

6.2 Orbit equivalence relations embedding E[subscript 0] 137

6.3 The Harrington-Kechris-Louveau theorem 141

6.4 Consequences of the Glimm-Effros dichotomy 147

6.5 Actions of cli Polish groups 151

7 Countable Borel Equivalence Relations 157

7.1 Generalities of countable Borel equivalence relations 157

7.2 Hyperfinite equivalence relations 160

7.3 Universal countable Borel equivalence relations 165

7.4 Amenable groups and amenable equivalence relations 168

7.5 Actions of locally compact Polish groups 174

8 Borel Equivalence Relations 179

8.1 Hypersmooth equivalence relations 179

8.2 Borel orbit equivalence relations 184

8.3 A jump operator for Borel equivalence relations 187

8.4 Examples of F[subscript sigma] equivalence relations 193

8.5 Examples of [Pi subscript 3 superscript 0] equivalence relations 196

9 Analytic Equivalence Relations 201

9.1 The Burgess trichotomy theorem 201

9.2 Definable reductions among analytic equivalence relations 206

9.3 Actions of standard Borel groups 210

9.4 Wild Polish groups 213

9.5 The topological Vaught conjecture 219

10 Turbulent Actions of Polish Groups 223

10.1 Homomorphisms and generic ergodicity 223

10.2 Local orbits of Polish group actions 227

10.3 Turbulent and generically turbulent actions 230

10.4 The Hjorth turbulence theorem 235

10.5 Examples of turbulence 239

10.6 Orbit equivalence relations and E[subscript 1] 241

III Countable Model Theory 245

11 Polish Topologies of Infinitary Logic 247

11.1 A review of first-order logic 247

11.2 Model theory of infinitary logic 252

11.3 Invariant Borel classes of countable models 256

11.4 Polish topologies generated by countable fragments 262

11.5 Atomic models and G[subscript delta] orbits 266

12 The Scott Analysis 273

12.1 Elements of the Scott analysis 273

12.2 Borel approximations of isomorphism relations 279

12.3 The Scott rank and computable ordinals 283

12.4 A topological variation of the Scott analysis 286

12.5 Sharp analysis of S[subscript infinity]-orbits 292

13 Natural Classes of Countable Models 299

13.1 Countable graphs 299

13.2 Countable trees 304

13.3 Countable linear orderings 310

13.4 Countable groups 314

IV Applications to Classification Problems 321

14 Classification by Example: Polish Metric Spaces 323

14.1 Standard Borel structures on hyperspaces 323

14.2 Classification versus nonclassification 329

14.3 Measurement of complexity 334

14.4 Classification notions 339

15 Summary of Benchmark Equivalence Relations 345

15.1 Classification problems up to essential countability 345

15.2 A roadmap of Borel equivalence relations 348

15.3 Orbit equivalence relations 350

15.4 General [Sigma subscript 1 superscript 1] equivalence relations 352

15.5 Beyond analyticity 353

A Proofs about the Gandy-Harrington Topology 355

A.1 The Gandy basis theorem 355

A.2 The Gandy-Harrington topology on X[subscript low] 358.

Notes:

"A Chapman & Hall book."

Includes bibliographical references (pages 361-372) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Hazel M. Hussong Fund.

ISBN:

9781584887935

1584887931

OCLC:

150375706